Vertex sliding along line until intersection
3 options to create a vertex at the intersection between two edges:


Activate and use The tinyCAD shipped add on > V2X: 'Vertex at Intersection'
EExtrude the edges to make faces, select them, and search for 'Intersect' (Spacebar in 2.79, F3 in 2.80). Check the 'Self Intersect' option. Delete excess vertices. 





Create a plane below the edges, and Knife Project the edges down onto the plane, which will create a vertex at the intersection between the projected edges, (and possibly some surplus geometry) That vertex can become a snap target, or you can duplicate the edges out of the plane.




This last method can sometimes be quite useful ..  the lines only have to intersect in your chosen projection, which you define by the orientation of your plane, before  Shift Numpad 7 aligning your view to it, to execute the Knife Project.